# Radians per Second to RPM Converter > Convert angular velocity from radians per second (rad/s) to revolutions per minute (RPM) and vice versa **Category:** Conversion **Keywords:** rad/s, rpm, radian per second, revolution per minute, angular velocity, conversion **URL:** https://complete.tools/rad-s-to-rpm-converter ## How it calculates The conversion between radians per second and RPM is based on the fundamental relationship between radians and revolutions. One complete revolution equals 2π radians (approximately 6.283185 radians), and one minute contains 60 seconds. Using these relationships, the formulas are derived as follows: To convert rad/s to RPM: RPM = rad/s × 60 / (2π) = rad/s × 30/π ≈ rad/s × 9.5493 To convert RPM to rad/s: rad/s = RPM × 2π / 60 = RPM × π/30 ≈ RPM × 0.10472 For example, if a motor shaft rotates at 100 rad/s, the equivalent speed in RPM would be: 100 × 9.5493 = 954.93 RPM. Conversely, if an engine runs at 3000 RPM, the angular velocity in rad/s would be: 3000 × 0.10472 = 314.16 rad/s. The tool also calculates hertz (Hz) as revolutions per second, which equals RPM divided by 60 or rad/s divided by 2π. Degrees per second is calculated by multiplying rad/s by 180/π (approximately 57.2958). ## Who should use this Mechanical engineers frequently use this tool when designing rotating machinery, analyzing motor specifications, or calculating gear ratios. Understanding the relationship between rad/s and RPM is essential for specifying motor speeds, calculating torque requirements, and ensuring mechanical components operate within safe limits. Physics students and educators benefit from this converter when studying rotational dynamics, angular momentum, and circular motion. The SI unit rad/s is standard in physics equations, while practical applications often express speeds in RPM. Automotive technicians and enthusiasts use this tool to convert between engine RPM readings from tachometers and the rad/s values needed for calculating wheel speeds, drivetrain dynamics, and centripetal forces on rotating components. Robotics engineers require angular velocity conversions when programming servo motors, calculating joint velocities, and ensuring proper synchronization between multiple rotating actuators in robotic systems. Industrial maintenance professionals use this converter when working with equipment specifications that may list speeds in different units, such as comparing international machinery ratings or interpreting sensor data from various systems. ## How to use To use this converter, first select the conversion direction using the segmented control at the top. Choose "rad/s to RPM" if you have a value in radians per second that you want to convert, or "RPM to rad/s" if you have a rotational speed in revolutions per minute. Enter your numeric value in the input field. The tool accepts decimal values for precision, such as 3.14159 rad/s or 1750.5 RPM. After entering your value, click the Convert button to see the results. The hero result card displays the primary converted value prominently, with a formula breakdown showing the calculation. Below this, you will find a results grid showing the angular velocity in multiple units: rad/s, RPM, hertz, and degrees per second. The speed gauge provides a visual representation of where the angular velocity falls on a scale from 0 to 10,000 RPM, with color coding indicating the speed category (low, medium, high, or very high speed). A reference table shows common angular velocities for everyday applications like hard drives, washing machines, car engines, and ceiling fans, helping you contextualize your converted values. ## Worked examples Example 1: A centrifuge operates at 628.32 rad/s. To find the RPM: 628.32 × 9.5493 = 6000 RPM. This indicates the centrifuge completes 6000 rotations per minute, which is typical for laboratory sample separation. Example 2: A car engine idles at 800 RPM. Converting to rad/s: 800 × 0.10472 = 83.78 rad/s. This value is useful for calculating the centripetal acceleration on engine components or the angular momentum of the crankshaft. Example 3: A DC motor is rated for 104.72 rad/s. Converting to RPM: 104.72 × 9.5493 = 1000 RPM. The motor completes 1000 revolutions per minute, and this also equals 16.67 Hz (revolutions per second). Example 4: A hard drive spins at 7200 RPM. The angular velocity in rad/s is: 7200 × 0.10472 = 754 rad/s. In degrees per second, this equals approximately 43,200 °/s, meaning the disk completes 120 full rotations every second. Example 5: A wind turbine blade rotates at 2.094 rad/s. Converting to RPM: 2.094 × 9.5493 = 20 RPM. Large wind turbines typically rotate slowly to maintain structural integrity while still generating significant power through high torque. ## Limitations This tool performs mathematical conversions and does not account for physical constraints of rotating systems. Actual motor speeds may vary due to load, temperature, voltage fluctuations, and mechanical losses not reflected in the conversion. The precision of results depends on the precision of input values. While the tool maintains high decimal accuracy internally, very large or very small numbers may display in scientific notation for readability. The speed categories (low, medium, high, very high) are general guidelines based on typical industrial applications. Different industries and applications may have different speed thresholds for what constitutes safe or normal operation. The tool assumes steady-state rotation and does not account for acceleration, deceleration, or variable speed profiles that occur in real-world applications during startup, shutdown, or speed changes. Negative values are accepted mathematically to represent reverse rotation, but physical interpretation of negative angular velocity depends on the reference frame and coordinate system used in your specific application. ## FAQs **Q:** Why are there two different units for angular velocity? **A:** Radians per second (rad/s) is the SI standard unit used in physics and engineering calculations because radians are dimensionless and simplify many formulas involving rotation. RPM is used in practical applications because it directly indicates how many complete rotations occur per minute, which is intuitive for machinery operators and technicians. **Q:** What does the speed category classification mean? **A:** The speed categories provide context for the rotational speed. Low speed (under 1800 RPM) includes applications like fans and slow motors. Medium speed (1800-3600 RPM) covers many industrial motors. High speed (3600-7200 RPM) includes high-performance motors and hard drives. Very high speed (above 7200 RPM) includes centrifuges and specialized equipment. **Q:** How is hertz related to RPM and rad/s? **A:** Hertz (Hz) measures frequency in cycles per second. Since one cycle equals one revolution, 1 Hz equals 60 RPM (60 revolutions per minute). In rad/s, 1 Hz equals 2π rad/s (approximately 6.283 rad/s) because one revolution spans 2π radians. **Q:** Can I use this for calculating centripetal force? **A:** Yes, once you have the angular velocity in rad/s, you can calculate centripetal acceleration using a = ω²r, where ω is the angular velocity in rad/s and r is the radius. Then multiply by mass for centripetal force: F = mω²r. **Q:** What is the relationship between angular velocity and linear velocity? **A:** Linear velocity (v) equals angular velocity (ω) multiplied by the radius (r): v = ωr. This means a point farther from the center of rotation travels faster in linear terms, even though all points complete the rotation in the same time. **Q:** Why is 2π used in the conversion formulas? **A:** The number 2π (approximately 6.283185) represents the number of radians in one complete circle. Since one revolution traces a complete circle, and a circle contains 2π radians, this constant bridges the relationship between revolutions and radians. --- *Generated from [complete.tools/rad-s-to-rpm-converter](https://complete.tools/rad-s-to-rpm-converter)*